4 Climate Damages

We assume that capital, output, both investments, and consumption are all diminished proportionately by \(N_t\). Our damage specification uses a piecewise log quadratic specification as a function of the temperature anomaly \(y.\) We suppose that the derivative of the logarithm of damages \(\hat{n}\) with respect to the temperature anomaly is

\[\begin{split}\begin{equation} \begin{array} {llll} \frac {d {\hat n}} {dy} & = \lambda_1 + \lambda_2 y & y \le {\tilde y} \\ \frac {d {\hat n}} {d{y}} & = \lambda_1 + \lambda_2 \left( y - {\tilde y} + {\bar y} \right) + \lambda_3(z_n ) (y - {\tilde y}) & y > {\tilde y} \end{array} \end{equation}\end{split}\]

for \(z_n \in \left\{ 1, 2, ..., L_n \right\}\).

This equation has an initial condition \({\hat n}(0) = 0.\) In the stochastic version of what follows, \({\tilde y}\) will be triggered by a Poisson jump prior to a temperature threshold \({\bar y}.\) We specify the intensity so that this jump takes place in the interval \([{\underline y}, {\bar y}].\) We shift the derivative of damages with respect to temperature to the right as captured by the change from \(\lambda_2 y\) to \(\lambda_2 \bar{y}\). We also increase the slope by including a term \(\lambda_3(z_n) ( y - {\tilde y})\), where the coefficient \(\lambda_3(z_n)\) is ex ante uncertain.

We plot the implied damage functions for thresholds between \({\tilde y} = 1.5\) and \({\tilde y} = 2.\), including a range of \(\lambda_3\)’s used in our quantitative policy assessment.

from src.plot import plot_damage
plot_damage("""Figure 3: Range of Possible Damage Functions for Different Jump Thresholds""")